2 The Homogeneity Assumption OLS assumes all cases in your data are comparablex’s are a sample drawn from a single populationBut we may analyze distinct groups of cases together in one analysisMean value of y may differ by group
3 Qualitative Variables These group effects remain as part of the error termIf groups differ in their distribution of x’s, then we get a correlation between the X variables and the error termViolates assumption: cov(Xi, ui)=E(u)=0Omitted Variable Bias!
4 Testing for Differences Across Groups (p. 249-252) The Chow Test:Is only valid under homoskedasticity (the error variance for the two groups must be equal).The null hypothesis is that there is no difference at all; either in the intercept or the slope between the two groups.This may be two restrictive in these cases, we should allow dummy variables and dummy interactions to allow us to predict different slopes and intercepts for the two groups.The Chow Testi.e. Testing for difference between males and females on academic performance.SSR1=Males only; SSR2=Females onlySSRur=SSR1+SSR2SSRP=SSRr=Pooling across both groups
5 Example: Democracy & Tariffs But if Democracies are more likely to be in RTA’s, then pooling RTA and non-RTA states biases the coefficientHere we see that democracies have lower tariffsHere we see that states in Regional Trading Arrangements (RTA’s) have lower tariffs
6 Solution: The Qualitative Variable Measure this group difference (RTA vs. Non-RTA) and specify it as an xThis eliminates biasBut we have no numerical scale to measure RTA’sCreate a categorical variable that captures this group difference
7 The Qualitative “Dummy” Create a variable that equals 1 when a case is part of a group, 0 otherwiseThis variable creates a new intercept for the cases in the group marked by the dummySpecifically, how would we interpret:
9 Graphical Depiction of a Dummy x1 (could be continuous, categorical, or dichotomous)
10 Multiple Category Dummies Dummy variables are a very flexible way to assess categorical differences in the mean of yWe can use dummies even for concepts with multiple categoriesImagine we want to capture the impact of global region on tariffsRegions: Americas, Europe, Asia, Africa
11 Warning! Do not fall into the dummy variable trap! When you have entered both values of a dummy variable in the same regression. These two variables are linearly dependent. One will drop out.
12 Multiple Category Dummies Create 4 separate dummy variables - 1 for each regionInclude all except one of these dummies in the equationIf you include all 4 dummies you get perfect collinearity with the constant. The fourth dummy will drop out.Americas+Europe+Asia+Africa=1
13 Interpreting Multi-Category Dummies Each coefficient compares the mean for that group to the mean in the excluded categoryThus if:βhat2-βhat4 compare the mean tariff in each region to the mean in the AmericasMean in Americas is βhat0An alternative strategy is to drop the constant and run all dummies, as discussed last week.
14 Dumb DummiesDummy variables are easy, flexible ways to measure categorical conceptsThey CAN be just labels for ignoranceTry to use dummies to capture theoretical constructs not empirical observationsIf possible, measure the theoretical construct more directly
15 Interaction Effects Dummy variables specify new intercepts Other slope coefficients in the equation do not changeOLS assumes that the slopes of continuous variables are constant across all casesWhat if slopes are different for different groups in our sample?
16 Interaction Effects: An Example What if the effect of democracy on tariffs depends on whether the state is in an RTA?
17 Interaction Effects: An Illustration (Notice that democracy has been converted to a dummy as well for illustration purposes)
18 How Do We Estimate This Set of Relationships? We begin with:Substituting for Βhat1, we get:Βhat1Βhat2Βhat3In STATA, they will appear as regular coefficients
25 Graphical Depiction of a Dummy/Continuous Interaction x1 (could be continuous, categorical, or dichotomous)
26 What if a Variable Interacts with Itself? What if Βhat1 depends on the value of x1?Then we substitute in as before:Curvilinear (Quadratic) effect is a type of interaction
27 More Complex Interactions We can use this method to specify the functional form of βhat1 in any way we chooseSimply substitute the function in for βhat1 , multiply out the terms and estimateOnly limitations are theories of interaction and levels of collinearity
28 Examples of interaction effects from my own research